Let a(x) and r(x) be positive continuous functions defined on the interval [0,∞), and let x→∞liminf(x−r(x))>0. Assume that y(x) is a continuous function on the whole real line, that it is differentiable on [0,∞), and that it satisfies y′(x)=a(x)y(x−r(x)) on [0,∞). Prove that the limit x→∞limy(x)exp{−∫0xa(u)du} exists and is finite.
I. Gyori